02_Single_Particle_Beamline.ipynb

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    "## Tracking a single particle in a beamline.\n",
    "\n",
    "These exercises are meant to get familiar with how to construct simple beamlines and to track single particles."
   ]
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   "source": [
    "#### Setup of your Python kernel\n",
    "Let's load our standard library functions, and setup matplotlib for nice inline plots"
   ]
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    "# Import custom tracking functions and useful libraries\n",
    "from tracking_library import *\n",
    "\n",
    "# Set up plotting\n",
    "%matplotlib inline\n",
    "params = {'legend.fontsize': 'x-large',\n",
    "          'figure.figsize': (15, 5),\n",
    "         'axes.labelsize': 'x-large',\n",
    "         'axes.titlesize':'x-large',\n",
    "         'xtick.labelsize':'x-large',\n",
    "         'ytick.labelsize':'x-large'}\n",
    "plt.rcParams.update(params)"
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    "### Exercise 6\n",
    "Recall that the **imaging equation** for a lens is $1/b+1/g=1/f,$ which corresponds to a system of one focusing lens with focal length $f,$ sandwiched between drift spaces with lengths $b$ and $g$.\n",
    "\n",
    "1. Write a beamline description that corresponds to this system, with, for example, $b=3$ m and $g=2$ m.\n",
    "2. Prove that the particles starting with a random angle $x'_0$ and zero position $x$ are focused back to the end of such a system.\n",
    "   \n",
    "   **Hint:** you can create a random ensample of particles with such properties as\n",
    "   \n",
    "   `X = np.array([np.zeros(n_particles),np.random.randn(n_particles)])`\n",
    "\n",
    "   where `n_particles` is the numbers of particles you want to generate.\n",
    "\n",
    "4. (Optional) Try to change the position of the quadrupole, i.e. $b$, while keeping fixed the total length of the imaging system (i.e. $b+g=5$ m). Plot both position and angle trajectory for one particle with $x=0$ and $x'_0=1$ mrad for different values of $b$, and show that, with an approriate change of $f$, this particle is always focused back to $x=0$ at the end of system, but what happens to the angle?\n",
    "\n",
    "   **Hint:** for this exercise it is useful to make a small animation with the `interactive` package. "
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   "source": [
    "#### SOLUTION 1"
   ]
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   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "#### SOLUTION 2"
   ]
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   "outputs": [],
   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "#### SOLUTION 3\n"
   ]
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   "outputs": [],
   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
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   "source": [
    "### Exercise 7\n",
    "\n",
    "Define a **FODO beamline** that **starts in the middle of a drift**. \n",
    "Then, model a particle that is on the optical axis, but has an initial angle $x' = 1$ mrad and plot the position $x$ along the beam line.\n",
    "For the moment, we can define arbitrarily the $f$ and $L$ parameters: we chose 2.5 m and 1 m, respectively. \n",
    "\n",
    "**Hint 1:** A FODO cell is one of the most fundamental lattice topologies. Look back at [Wolfgan's lecture](https://indico.cern.ch/event/1117526/contributions/4978192/). \n",
    "As in [Wolfgan's lecture](https://indico.cern.ch/event/1117526/contributions/4978192/), we define $L$ as the length of one drift, i.e. the total length of the cell ($L_{FODO}$) is $2L$. \n",
    "\n",
    "<p align=\"center\">\n",
    "<img src=./_img_exercises/slide_fodo.png width=\"60%\">\n",
    "</p>\n",
    "\n",
    "**Hint 2:** To better visualize the trajectory of the particle(s), always split the drifts in several shorter drifts (e.g. $10$ drifts, each $L/10$ long)."
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   "source": [
    "#### SOLUTION \n",
    "The first part of the exercise is to define a FODO cell structure.  In its simpler version, it consists on two thin quadrupoles of opposite focal length, f, spaced by two drifts of length L (2*L is the total length of the cell). \n",
    "\n",
    "We could start the FODO from an arbitrary point inside it, but we decided to start describing it from the center of a drift.\n",
    "\n",
    "We also suggest to split the drift in small drifts, say 0.1 m long each."
   ]
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   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "### Exercise 8\n",
    "Plot the angle $x'$ along the beam line."
   ]
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   "source": [
    "#### SOLUTION"
   ]
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   "metadata": {},
   "outputs": [],
   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "### Exercise 9\n",
    "\n",
    "Plot both the position $x$ and the angle $x'$ through five cells.\n",
    "\n",
    "**Hint:** After having plotted $x$ trajectory on a plot, one can create a second vertical axis that shares the same horizontal axis with `plt.twinx()`"
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   "source": [
    "#### SOLUTION"
   ]
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   "metadata": {},
   "outputs": [],
   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "### Exercise 10\n",
    "\n",
    "Plot the position $x$ and angle $x'$ through 100 cells, play with different values of the focal length $f$ and explore whether you can make the oscillations grow.\n",
    "\n",
    "**Hint:** Look back at [Wolfgan's lecture](https://indico.cern.ch/event/1117526/contributions/4978192/):\n",
    "\n",
    "<p align=\"center\">\n",
    "<img src=./_img_exercises/slide_stability.png width=\"60%\">\n",
    "</p>"
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   "source": [
    "#### SOLUTION"
   ]
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   "source": [
    "# code here your solution...\n",
    "\n",
    "\n",
    "\n",
    "\n"
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   "source": [
    "It it important to observe that when the focal length approaches the stability limit (only for $f>L_{FODO}/4$ the machine is stable), then the oscillation start to grow during the transport in the lattice."
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